mth 158
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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My response to your question to one of my submissions.
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In one of my submissions, I didn’t understand a problem and you wrote:
Can you take this paragraph by paragraph, at least for the first few paragraphs, and tell me specifically what you do and do not understand about each step?
Here is my attempt at describing what I do and do not understand about each step in this problem:
Question: * 3.5.58 (was 3.4.40). h(x) = 4 / x + 2.
What basic function did you start with and, in order, what transformations were required to obtain the graph of the given function?
Describe your graph.
Give three points on your graph and tell to which basic points on the graph of your basic function each corresponds.
Your solution:
I know this is a reciprocal function, but I am not sure how to start.
set y = 4/x
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Given Solution:
We start with the basic reciprocal function y = 1 / x, which has vertical asymptote. At the y axis and horizontal asymptotes at the right and left along the x axis and passes through the points (-1, -1) and (1, 1).
First when I read this line, I had to actually look up the word asymptote because I needed to understand the meaning a little better to try and see the graph it was describing. Here is the definition I got on line…“A line whose distance to a given curve tends to zero. An asymptote may or may not intersect its associated curve”
I just checked the text and the author doesn't mention the word 'asymptote' until the 4th chapter, though he introduces the reciprocal function in the present chapter. That's a little unusual, and I'm not sure why the author didn't include the terminology but I expect he was trying to give a brief introduction to the function before further exploring rational functions in the next chapter (I think very highly of the author and his texts and respect his judgement).
To get y = 4 / x we must multiply y = 1 / x by 4. Multiplying a function by 4 results in a vertical stretch by factor 4, moving every point 4 times as far from the x axis.
Here, when you say multiplying a function by 4, are you saying you would do this:
1/x * 4/1 and if so, wouldn’t that give you 4/x then would you substitute the x = 1 and x = -1 to get the next set of points (-1, -4) and (1, 4)?
That is so.
However note that (1, 4) is 4 times as far from the x axis as the point (1, 1), and (-1, -4) is 4 times as far from the x axis as (-1, -1).
Given a graph of y = 1 / x, we could have simply taken each point of the graph and moved it 4 times as far from the x axis, without ever calculating y = 4 / x. We might well want to calculate the values, but it's important to visualize, without the point-by-point calculations, how the graphs transform.
At this point (if I am correct), I can understand how to get the -4, and 4. Does the x value always remain 1 and -1 in this particular situation?
We use the points at x = -1 and x = 1 as reference points for the graph. Take another look at this problem (which is in query 25) and you'll see an expanded explanation with some graphs, which should answer this and a number of other questions for you.
This will not affect the location of the vertical or horizontal asymptotes but for example the points (-1, -1) and (1, 1) will be transformed to (-1, -4) and (1, 4). At every point the new graph will at this point be 4 times as far from the x axis.
So here, you are saying the next set of points for y = 4/x would be (-1, -8) and (1, 8) if we were to continue in this graph?
The newly posted query should be helpful here. The graph points at x = -1 and x = 1 are still (-1, -4) and (1, 4). The points (-1, -8) and (1, 8) do not lie on this graph. The points (2, 8) and (-2, -8) do, but they aren't particularly important for the transformation picture we're trying to develop.
At this point we have the graph of y = 4 / x. The function we wish to obtain is y = 4 / x + 2.
Adding 2 in this manner increases the y value of each point by 2. The point (-1, -4) will therefore become (-1, -4 + 2) = (-1, -2). The point (1, 4) will similarly be raised to (1, 6).
So, by adding the 2, the original points (-1, -4) and (1, 4) were just used to show how 4/x would look, and then by adding the y = 4/x + 2 is the “real” graph we are trying to obtain?
That's the right idea.
Since every point is raised vertically by 2 units, with no horizontal shift, the y axis will remain an asymptote. As the y = 4 / x graph approaches the x axis, where y = 0, the y = 4 / x + 2 graph will approach the line y = 0 + 2 = 2, so the horizontal asymptotes to the right and left will consist of the line y = 2.
This paragraph is the most confusing. Up to this point (if my assumptions were correct) I could follow along, but the wording is a bit confusing. For instance: the y axis will remain an asymptote….I understand…as the y = 4/x graph approaches the x axis….I understand that……and then it says where y = 0, the y = 4/x + 2 graph will approach the line y = 0 + 2 = 2……throws me off. I don’t understand this part…..and the horizontal asymptotes to the right and left will consist of the line y = 2…I tried drawing the graph and looking at it, but it doesn’t appear that anything = 2.
Our final graph will have asymptotes at the y axis and at the line y = 2, with basic points (-1, -2) and (1, 6).
Naturally, if I didn’t understand the prior paragraph, I don’t understand this one for the same reasons.
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Self-critique (if necessary):
I am having trouble trying to grasp this one even after reading the given solution.
Can you take this paragraph by paragraph, at least for the first few paragraphs, and tell me specifically what you do and do not understand about each step?
These are great questions, and this is a key problem with some key concepts. I've done an extensive edit on that solution in Query 25. Check it out, along with my notes here. I'll be glad to answer further questions.